Let $M \in \mathbb{R}^{d\times d}$ be an antisymmetric matrix. Is there a lower/upper bound or equality relating the two quantities $$ \min_{u \in \mathbb{C}^d, \lVert u \rVert = 1} \left|u^*Au\right|^2 \qquad \text{and} \qquad \min_{u \in \mathbb{C}^d, \lVert u \rVert = 1} u^*A^TA u \, ?$$ The right-hand side is the square of the smallest singular value of $A$. Also notice that $u^* A u$ must be pure imaginary while $u^* A^T A u$ must be real.
Indeed, the comment below by Stephen shows that the left-hand side is zero. What about general matrices $A$, not necessarily antisymmetric?
Thanks Stephen for pointing to the Cauchy-Scharz inequality: we have $$ \left| u^* A u \right|^2 = \left| \left< Au, u \right> \right|^2 \leq \left< Au, Au \right> \left< u, u \right> = u^* A^T A u $$ for normal vector $u$ and real matrix $A$, hence $$ \min_{u \in \mathbb{C}^d, \lVert u \rVert = 1} \left|u^*Au\right|^2 \leq \min_{u \in \mathbb{C}^d, \lVert u \rVert = 1} u^*A^TA u $$ for any real matrix $A$. The left-hand side is zero for antisymmetric $A$.